I'm studying some group cohomology, and I'm stuck on the following problem (namely, problem VIII.1.3 of Brown's Cohomology of Groups): let's define the geometric dimension of a group $G$ as $$ \text{gd}G = \inf\{n \, | \, \dim_{CW}(X), \, X \text{ a classifying space for }G \}, $$ where we take as classifying spaces the $K(G,1)$'s that are also path-connected CW-complexes.
Similarly to the case of cohomological dimension (where the cohomological dimension of $G$ is defined as the infimum of the length of projective resolutions of $\mathbb{Z}$ over $\mathbb{Z} G$), I would like to show the following:
- if $A \subset G$ is a group inclusion, then $$ \text{gd} A \leq \text{gd} G; $$
- If $1 \to G_1 \to G \to G_2 \to 1$ is a group extension, then $$ \text{gd} G \leq \text{gd} G_1 + \text{gd} G_2; $$
- If $G = G_1 \ast_A G_2$ where $A \subset G_1, G_2$, then $$ \text{gd} G \leq \max \{\text{gd} G_1, \text{gd} G_2, 1 + \text{gd} A\}. $$
My attempt:
For 1: this is more or less clear. Any classifying space $X$ as above admits an universal cover $\tilde{X} \to X$; by the classification theorem of connected covers of $X$, we get another cover $p_A: Y \to X$ such that ${p_{A}}_\ast(\pi_1(Y)) = A \subset G = \pi_1(X)$, and the induced map $\tilde{X} \to Y$ is also a cover.
I claim that $Y$ is a classifying space: first of all, since ${p_A}_\ast$ is injective, $\pi_1(Y) \cong A$. Moreover, $\tilde{X}$ is a CW complex (by lifting the CW-structure from $X$) such that $\dim_{CW}(\tilde{X}) = \dim_{CW}(X)$, and $\pi_{\geq 2}(\tilde{X}) = \pi_{\geq 2}(X) = 0$; by the same reasoning $Y$ is a CW-complex, $\pi_{\geq 2}(\tilde{Y}) = \pi_{\geq 2}(Y) = 0$ and we have that $\dim(Y) \leq \dim(X)$.
For 2: here I'm stuck. I know I can get a fibre sequence $K(G_1,1) \to K(G,1) \to K(G_2,1)$, and then use the Serre spectral sequence to get, for any abelian group $M$, $$ H^p(K(G_2,1);H^q(K(G_1,1);M)) \Rightarrow H^{p+q}(K(G,1);M), $$ and for $k> \text{gd}G_2 + \text{gd} G_1$, by taking $p+q =k$ and by the vanishing of the second page, I get that $H^{k}(K(G,1); M) \cong 0$. Is this enough to conclude? If yes, how?
For 3: also here I'm stuck, similarly as before. Briefly, the idea I have is: take $X_1$ and $X_2$ classifying spaces for $G_1$ and $G_2$ respectively, minimizing the dimensions. Since $A \subset G_1, G_2$ and $\text{Hom}(A, G_i) \cong [K(A,1), K(G_i,1)]_\ast$, we can find maps from $X_A$, the $K(A,1)$ minimizing the dimension for $A$, fitting in a diagram of cellular maps: $$ X_1 \leftarrow X_A \rightarrow X_2 $$
By taking the pushout and calling it $X$, by van Kampen and Mayer-Vietoris (applied on the universal cover), this should be a classifying space for $G$. I guess that at this point (at least intuitively) $\text{gd} G \leq \max\{\text{gd} G_1, \text{gd} G_2 \}$, but I can't get the dimension of $A$ coming into play here (maybe I should use some sort of Mayer-Vietoris technique here)? Also, does this make sense, even though it's really sketchy?
Any help is appreciated. Thanks!